Efficient Domination and Polarity
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A Characterization of General Position Sets in Graphs
A characterization of general position sets in graphs Bijo S. Anand a Ullas Chandran S. V. b Manoj Changat c Sandi Klavˇzar d;e;f Elias John Thomas g November 16, 2018 a Department of Mathematics, Sree Narayana College, Punalur-691305, Kerala, India; bijos [email protected] b Department of Mathematics, Mahatma Gandhi College, Kesavadasapuram, Thiruvananthapuram-695004, Kerala, India; [email protected] c Department of Futures Studies, University of Kerala Thiruvananthapuram-695034, Kerala, India; [email protected] d Faculty of Mathematics and Physics, University of Ljubljana, Slovenia [email protected] e Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia f Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia g Department of Mathematics, Mar Ivanios College, Thiruvananthapuram-695015, Kerala, India; [email protected] Abstract A vertex subset S of a graph G is a general position set of G if no vertex of S lies on a geodesic between two other vertices of S. The cardinality of a largest general position set of G is the general position number gp(G) of G. It is proved that S ⊆ V (G) is a general position set if and only if the components of G[S] are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of S. If diam(G) = 2, then gp(G) is the maximum of the clique number of G and the maximum order of an induced complete multipartite subgraph of the complement of G. As a consequence, gp(G) of a cograph G can be determined in polynomial time. -
Diestel: Graph Theory
Reinhard Diestel Graph Theory Electronic Edition 2000 c Springer-Verlag New York 1997, 2000 This is an electronic version of the second (2000) edition of the above Springer book, from their series Graduate Texts in Mathematics, vol. 173. The cross-references in the text and in the margins are active links: click on them to be taken to the appropriate page. The printed edition of this book can be ordered from your bookseller, or electronically from Springer through the Web sites referred to below. Softcover $34.95, ISBN 0-387-98976-5 Hardcover $69.95, ISBN 0-387-95014-1 Further information (reviews, errata, free copies for lecturers etc.) and electronic order forms can be found on http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ http://www.springer-ny.com/supplements/diestel/ Preface Almost two decades have passed since the appearance of those graph the- ory texts that still set the agenda for most introductory courses taught today. The canon created by those books has helped to identify some main elds of study and research, and will doubtless continue to inuence the development of the discipline for some time to come. Yet much has happened in those 20 years, in graph theory no less than elsewhere: deep new theorems have been found, seemingly disparate methods and results have become interrelated, entire new branches have arisen. To name just a few such developments, one may think of how the new notion of list colouring has bridged the gulf between invari- ants such as average degree and chromatic number, how probabilistic methods and the regularity lemma have pervaded extremal graph theo- ry and Ramsey theory, or how the entirely new eld of graph minors and tree-decompositions has brought standard methods of surface topology to bear on long-standing algorithmic graph problems. -
Multipartite Graph Algorithms for the Analysis of Heterogeneous Data
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 12-2015 Multipartite Graph Algorithms for the Analysis of Heterogeneous Data Charles Alexander Phillips University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Computational Biology Commons Recommended Citation Phillips, Charles Alexander, "Multipartite Graph Algorithms for the Analysis of Heterogeneous Data. " PhD diss., University of Tennessee, 2015. https://trace.tennessee.edu/utk_graddiss/3600 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Charles Alexander Phillips entitled "Multipartite Graph Algorithms for the Analysis of Heterogeneous Data." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Computer Science. Michael A. Langston, Major Professor We have read this dissertation and recommend its acceptance: Bruce J. MacLennon, Brynn H. Voy, David J. Icove Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Multipartite Graph Algorithms for the Analysis of Heterogeneous Data A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Charles Alexander Phillips December 2015 Copyright © 2015 by Charles A. -
Decomposition and Domination of Some Graphs Fairouz Beggas
Decomposition and Domination of Some Graphs Fairouz Beggas To cite this version: Fairouz Beggas. Decomposition and Domination of Some Graphs. Data Structures and Algorithms [cs.DS]. Université Claude Bernard Lyon 1, 2017. English. tel-02168197 HAL Id: tel-02168197 https://hal.archives-ouvertes.fr/tel-02168197 Submitted on 28 Jun 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No d'ordre NNT : 2017LYSE1051 THESE` DE DOCTORAT DE L'UNIVERSITE´ DE LYON op´er´eeau sein de l'Universit´eClaude Bernard Lyon 1 Ecole´ Doctorale ED 512 Informatique et Math´ematiques(InfoMaths) Sp´ecialit´ede doctorat : Informatique Soutenue publiquement le 28/03/2017, par : Fairouz BEGGAS Decomposition and Domination of Some Graphs Devant le jury compos´ede : Jean-Luc Baril, Professeur, Universit´ede Bourgogne Rapporteur Lhouari Nourine, Professeur, Universit´eBlaise Pascal Clermont-Ferrand Rapporteur Daniela Grigori, Professeur, Universit´ede Paris Dauphine Examinatrice Salima Benbernou, Professeur, Universit´ede Paris Descartes Examinatrice Norma Zagaglia Salvi, Professeur, Ecole´ polytechnique de Milan Examinatrice Hamamache Kheddouci, Professeur, Universit´eLyon 1 Directeur de th`ese Mohammed Haddad, Maitre de Conf´erences,Universit´eLyon 1 Co-directeur de th`ese iii Acknowledgments I would like to express my deepest gratitude to my advisor, Pr. -
Characterization of General Position Sets and Its Applications to Cographs and Bipartite Graphs
Characterization of general position sets and its applications to cographs and bipartite graphs Bijo S. Anand a Ullas Chandran S. V. b Manoj Changat c Sandi Klavˇzar d;e;f Elias John Thomas g April 16, 2019 a Department of Mathematics, Sree Narayana College, Punalur-691305, Kerala, India; bijos [email protected] b Department of Mathematics, Mahatma Gandhi College, Kesavadasapuram, Thiruvananthapuram-695004, Kerala, India; [email protected] c Department of Futures Studies, University of Kerala Thiruvananthapuram-695034, Kerala, India; [email protected] d Faculty of Mathematics and Physics, University of Ljubljana, Slovenia [email protected] e Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia f Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia g Department of Mathematics, Mar Ivanios College, Thiruvananthapuram-695015, Kerala, India; [email protected] Abstract A vertex subset S of a graph G is a general position set of G if no vertex of S lies on a geodesic between two other vertices of S. The cardinality of a largest general position set of G is the general position number gp(G) of G. It is proved that S ⊆ V (G) is in general position if and only if the components of G[S] are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of S. If diam(G) = 2, then gp(G) is the maximum of !(G) and the maximum order of an induced complete multipartite subgraph of the complement of G. As a consequence, gp(G) of a cograph G can be determined in polynomial time. -
Characterizations of Cographs As Intersection Graphs of Paths on a Grid
Discrete Applied Mathematics 178 (2014) 46–57 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Characterizations of cographs as intersection graphs of paths on a grid Elad Cohen a,∗, Martin Charles Golumbic a, Bernard Ries b,c a Caesarea Rothschild Institute and Department of Computer Science, University of Haifa, Mount Carmel, Haifa 31905, Israel b PSL, Université Paris-Dauphine, 75775 Paris Cedex 16, France c CNRS, LAMSADE UMR 7243, France article info a b s t r a c t Article history: A cograph is a graph which does not contain any induced path on four vertices. In this Received 20 June 2013 paper, we characterize those cographs that are intersection graphs of paths on a grid in the Received in revised form 18 June 2014 following two cases: (i) the paths on the grid all have at most one bend and the intersections Accepted 25 June 2014 concern edges (! B -EPG); (ii) the paths on the grid are not bended and the intersections Available online 26 July 2014 1 concern vertices (! B0-VPG). In both cases, we give a characterization by a family of forbidden induced subgraphs. We Keywords: further present linear-time algorithms to recognize B -EPG cographs and B -VPG cographs Cograph 1 0 Cotree using their cotree. Grid ' 2014 Elsevier B.V. All rights reserved. Intersection graph Induced subgraph 1. Introduction Edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. -
Graph Theory, an Antiprism Graph Is a Graph That Has One of the Antiprisms As Its Skeleton
Graph families From Wikipedia, the free encyclopedia Chapter 1 Antiprism graph In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3- vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.[1] 1.1 Examples The first graph in the sequence, the octahedral graph, has 6 vertices and 12 edges. Later graphs in the sequence may be named after the type of antiprism they correspond to: • Octahedral graph – 6 vertices, 12 edges • square antiprismatic graph – 8 vertices, 16 edges • Pentagonal antiprismatic graph – 10 vertices, 20 edges • Hexagonal antiprismatic graph – 12 vertices, 24 edges • Heptagonal antiprismatic graph – 14 vertices, 28 edges • Octagonal antiprismatic graph– 16 vertices, 32 edges • ... Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting) antiprisms, the star antiprisms, they do not form a different sequence of graphs. 1.2 Related graphs An antiprism graph is a special case of a circulant graph, Ci₂n(2,1). Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs. 1.3 References [1] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270. 2 1.4. EXTERNAL LINKS 3 1.4 External links • Weisstein, Eric W., “Antiprism graph”, MathWorld.